Strict: BoolElt                     Default: false

    Exception: DivFunElt                Default: false

    Raw: BoolElt                        Default: false

    Strict: BoolElt                     Default: false

    IntegralOutside: BoolElt            Default: false

Example FldFunAb_strong-approximation (H47E7)

We first have to define a function field and some places:

> k<w> := GF(4);
> kt<t> := PolynomialRing(k);
> ktx<x> := PolynomialRing(kt);
> K := FunctionField(x^3-w*t*x^2+x+t);
> lp := Places(K, 2);

We will now try to find an element x in K such that v_pi(x - e_i) ≥m_i for p_i = lp[i], m_i = i and random elements e_i:

> e := [Random(K, 3) : i in lp];
> m := [i : i in [1..#lp]];
> D := &+ [ m[i]*lp[i] : i in [1..#lp]];
> x := StrongApproximation(D, [<lp[i], e[i]> : i in [1..#lp]]);
> [Valuation(x-e[i], lp[i]) : i in [1..#lp]];
[ 1, 2, 3, 4, 5, 6 ]

Note, that we only required ≥ for the valuations, to enforce = we would need to pass the Strict option. This will double the running time.

---

NonSpecialDivisor(m): DivFunElt -> DivFunElt, RngIntElt

    Exception: DivFunElt                Default: 

Given an effective divisor m, find a place P coprime to m and an integer r≥0 such that rP - m is a non special divisor and return r and P.

If Exception is specified, it must be an effective divisor n coprime to m. In this case the function finds r>0 such that rn - m is non special and returns r and n.

NormGroup(F) : FldFun -> DivFunElt, GrpAb

    Cond: DivFunElt                     Default: 

    AS: RngWittElt                      Default: 

    Extra: RngIntElt                    Default: 5

Given a global function field, try to compute its norm group. The norm group is defined to be the group generated by norms of unramified divisors. This group can be related to a subgroup of some ray class group.

Provided F is abelian, this function will compute a divisor m and a sub group U of the ray class group modulo m such that F is isomorphic to the ray class field thus defined.

This function uses a heuristic algorithm. It will terminate after the size of the quotient by the norm group is less or equal than the degree for Extra many places.

If Cond is given, it must be an effective divisor that will be used as the potential conductor of F. Note: if Cond is too small, ie. a proper divisor of the true conductor, the result of this function will be wrong. However, if the conductor is not passed in, the discriminant divisor is used as a starting point. As this is in general far too large, the function will be much quicker if a better (smaller) starting point is passed in.

If AS is given, it must be a Witt vector e of appropriate length and F should be the corresponding function field. This allows a much better initial guess for the conductor than using the discriminant.

Sign(a, p) : FldFunElt, PlcFunElt -> RngElt

Given a function a in some global function field and a place p such that a is integral at p (has non-negative valuation) return the sign of a, ie. the first non-zero coefficient if the expansion of a at p. The sign function is not unique. Magma choses a sign function when creating the residue class field map.

ChangeModel(F, p) : FldFun, PlcFunElt -> FldFun

Given a global function field F and a place p, return a new function field G that is F_q-isomorphic to F and has p as the only infinite place.

ArtinSchreierReduction(u, P): FldFunGElt, PlcFunElt -> RngIntElt, FldFunElt

Return the valuation of u - (z^p - z) at P and an element z such that this valuation is either positive or not congruent to 0 mod p where p is the characteristic of the field of u.

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